Googling the brain: discovering hierarchical and asymmetric network structures, with applications in neuroscience

Hierarchical organisation is a common feature of many directed networks arising in nature and technology. For example, a well-defined message-passing framework based on managerial status typically exists in a business organisation. However, in many real-world networks such patterns of hierarchy are unlikely to be quite so transparent. Due to the nature in which empirical data is collated the nodes will often be ordered so as to obscure any underlying structure. In addition, the possibility of even a small number of links violating any overall “chain of command” makes the determination of such structures extremely challenging. Here we address the issue of how to reorder a directed network in order to reveal this type of hierarchy. In doing so we also look at the task of quantifying the level of hierarchy, given a particular node ordering. We look at a variety of approaches. Using ideas from the graph Laplacian literature, we show that a relevant discrete optimization problem leads to a natural hierarchical node ranking. We also show that this ranking arises via a maximum likelihood problem associated with a new range-dependent hierarchical random graph model. This random graph insight allows us to compute a likelihood ratio that quantifies the overall tendency for a given network to be hierarchical. We also develop a generalization of this node ordering algorithm based on the combinatorics of directed walks. In passing, we note that Google’s PageRank algorithm tackles a closely related problem, and may also be motivated from a combinatoric, walk-counting viewpoint. We illustrate the performance of the resulting algorithms on synthetic network data, and on a real-world network from neuroscience where results may be validated biologically

Sustaining Glasgow's Urban Networks: the Link Communities of Complex Urban Systems

As cities grow in population size and became more crowded (UN DESA, 2018), the main future challenges around the world will remain to be accommodating the growing urban population while drastically reducing environmental pressure. Contemporary urban agglomerations (large or small) constantly impose burden on the natural environment by conveying ecosystem services to close and distant places, through coupled human nature [infrastructure] systems (CHANS). Tobler’s first law in geography (1970) that states that “everything is related to everything else, but near things are more related than distant things” is now challenged by globalization. When this law was first established, the hypothesis referred to geological processes (Campbell and Shin, 2012, p.194) that were predominantly observed in pre-globalized economy, where freight was costly and mainly localized (Zhang et al., 2018). With the recent advances and modernisation made in transport technologies, most of them in the sea and air transportation (Zhang et al., 2018) and the growth of cities in population, natural resources and bi-products now travel great distances to infiltrate cities (Neuman, 2006) and satisfy human demands. Technical modernisation and the global hyperconnectivity of human interactions and trading, in the last thirty years alone resulted with staggering 94 per cent growth of resource extraction and consumption (Giljum et al., 2015). Local geographies (Kennedy, Cuddihy and Engel-Yan, 2007) will remain affected by global urbanisation (Giljum et al., 2015), and as a corollary, the operational inefficiencies of their local infrastructure networks, will contribute even more to the issues of environmental unsustainability on a global scale. Another challenge for future city-regions is the equity of public infrastructure services and policy creation that promote the same (Neuman and Hull, 2009). Public infrastructure services refer to services provisioned by networked infrastructure, which are subject to both public obligation and market rules. Therefore, their accessibility to all citizens needs to be safeguarded. The disparity of growth between networked infrastructure and socio-economic dynamics affects the sustainable assimilation and equal access to infrastructure in various districts in cities, rendering it as a privilege. Yet, the empirical evidence of whether the place of residence acts as a disadvantage to public service access and use, remains rather scarce (Clifton et al., 2016). The European Union recognized (EU, 2011) the issue of equality in accessibility (i.e. equity) critical for territorial cohesion and sustainable development across districts, municipalities and regions with diverse economic performance. Territorial cohesion, formally incorporated into the Treaty of Lisbon, now steers the policy frameworks of territorial development within the Union. Subsequently, the European Union developed a policy paradigm guided by equal access (Clifton et al., 2016) to public infrastructure services, considering their accessibility as instrumental aspect in achieving territorial cohesion across and within its member states. A corollary of increasing the equity to public infrastructure services among growing global population is the potential increase in environmental pressure they can impose, especially if this pressure is not decentralised and surges at unsustainable rate (Neuman, 2006). This danger varies across countries and continents, and is directly linked to the increase of urban population due to; [1] improved quality of life and increased life expectancy and/or [2] urban in-migration of rural population and/or [3] global political or economic immigration. These three rising urban trends demand new approaches to reimagine planning and design practices that foster infrastructure equity, whilst delivering environmental justice. Therefore, this research explores in depth the nature of growth of networked infrastructure (Graham and Marvin, 2001) as a complex system and its disparity from the socio-economic growth (or decline) of Glasgow and Clyde Valley city-region. The results of this research gain new understanding in the potential of using emerging tools from network science for developing optimization strategy that supports more cecentralized, efficient, fair and (as an outcome) sustainable enlargement of urban infrastructure, to accommodate new and empower current residents of the city. Applying the novel link clustering community detection algorithm (Ahn et al., 2010) in this thesis I have presented the potential for better understanding the complexity behind the urban system of networked infrastructure, through discovering their overlapping communities. As I will show in the literature review (Chapter 2), the long standing tradition of centralised planning practice relying on zoning and infiltrating infrastructure, left us with urban settlements which are failing to respond to the environmental pressure and the socio-economic inequalities. Building on the myriad of knowledge from planners, geographers, sociologists and computer scientists, I developed a new element (i.e. link communities) within the theory of urban studies that defines cities as complex systems. After, I applied a method borrowed from the study of complex networks to unpack their basic elements. Knowing the link (i.e. functional, or overlapping) communities of metropolitan Glasgow enabled me to evaluate the current level of communities interconnectedness and reveal the gaps as well as the potentials for improving the studied system’s performance. The complex urban system in metropolitan Glasgow was represented by its networked infrastructure, which essentially was a system of distinct sub-systems, one of them mapped by a physical and the other one by a social graph. The conceptual framework for this methodological approach was formalised from the extensively reviewed literature and methods utilising network science tools to detect community structure in complex networks. The literature review led to constructing a hypothesis claiming that the efficiency of the physical network’s topology is achieved through optimizing the number of nodes with high betweenness centrality, while the efficiency of the logical network’s topology is achieved by optimizing the number of links with high edge betweenness. The conclusion from the literature review presented through the discourse on to the primal problem in 7.4.1, led to modelling the two network topologies as separate graphs. The bipartite graph of their primal syntax was mirrored to be symmetrical and converted to dual. From the dual syntax I measured the complete accessibility (i.e. betweenness centrality) of the entire area and not only of the streets. Betweenness centrality of a node measures the number of shortest paths that pass through the node connecting pairs of nodes. The betweenness centrality is same as the integration of streets in space syntax, where the streets are analysed in their dual syntax representation. Street integration is the number of intersections the street shares with other streets and a high value means high accessibility. Edges with high betweenness are shared between strong communities. Based on the theoretical underpinnings of the network’s modularity and community structure analysed herein, it can be concluded that a complex network that is both robust and efficient (and in urban planning terminology ‘sustainable’) is consisted of numerous strong communities connected with each other by optimal number of links with high edge betweenness. To get this insight, the study detected the edge cut-set and vertex cut-set of the complex network. The outcome was a statistical model developed in the open source software R (Ihaka and Gentleman, 1996). The model empirical detects the network’s overlapping communities, determining the current sustainability of its physical and logical topologies. Initially, an assumption was that the number of communities within the infrastructure (physical) network layer were different from the one in the logical. They were detected using the Louvain method that performs graph partitioning on the hierarchical streets structure. Further, the number of communities in the relational network layer (i.e. accessibility to locations) was detected based on the OD accessibility matrix established from the functional dependency between the household locations and predefined points of interest. The communities from the graph of the ‘relational layer' were discovered with the single-link hierarchical clustering algorithm. The number of communities observed in the physical and the logical topologies of the eight shires significantly deviated

The Web of Law

Scientists and mathematicians in recent years have become intensely interested in the structure of networks. Networks turn out to be crucial to understanding everything from physics and biology, to economics and sociology. This article proposes that the science of networks has important contributions to make to the study of law as well. Legal scholars have yet to study, or even recognize as such, one of the largest, most accessible, and best documented human-created networks in existence. This is the centuries-old network of case law and other legal authorities into which lawyers, judges, and legal scholars routinely delve in order to discover what the law is on any given topic. The network of American case law closely resembles the Web in structure. It has the peculiar mathematical and statistical properties that networks have. It can be studied using techniques that are now being used to describe many other networks, some found in nature, and others created by human action. Studying the legal network can shed light on how the legal system evolves, and many other questions. To initiate what I hope will become a fruitful new type of legal scholarship, I present in this article the preliminary results of a significant citation study of nearly four million American legal precedents, which was undertaken at my request by the LexisNexis corporation using their well-known Shepard\u27s citation service. This study demonstrates that the American case law network has the overall structure that network theory predicts it would. This article has three parts. First, I introduce some basic concepts of network science, including such important ideas as nodes, links, random graphs, evolving networks, scale-free networks, small worlds, the rich get richer dynamic, node fitness, and clusters. Oddly enough, the mathematical tools that have proven most useful for studying networks (or at least scale-free networks) come from statistical mechanics, a branch of physics. Having introduced network theory in Part I, and having presented evidence that American case law is a scale-free network in Part II, I argue for the significance of this discovery in Part III. I hope that by the time they reach Part III, readers will already be realizing the potential richness of applying network theory to legal systems. In Part III, I describe some insights that appear from this application and suggest areas for future research. The most famous hypothesis about the structure of law is that it is a seamless web. This old phrase, however, is just a metaphor we have used to grope for a reality we have not been in a position to express more precisely. Network science changes that. The Web of Law can be considered as a mathematical object whose topology can be analyzed using the tools pioneered by physicists and others who wanted to explore the structure of the Web and other real networks. The Web of Law has a structure very similar to that of other real networks, such as the Web and the network of scientific papers. The Web of Law is in substantial part a scale-free network, organized with hub cases that have many citations and the vast majority of cases, which have very few. The distribution of citation frequency approximates a power-law distribution, as is common with real scale-free networks, with truncations at either extreme of its distribution, which is also common. Many promising hypotheses can be generated by considering the law as a scale-free network. State and federal systems can be examined empirically to measure how well integrated each is with itself, and with each other, and how this is changing over time. Legal authorities can be measured to determine whether their authority is emerging or declining. Institutional bodies, such as courts, can be examined in the same way. Clusters of cases, which will reveal the semantic topology of law, can be mapped to determine whether traditional legal categories are accurate or require reform. These methods can be used to develop computer programs to improve the efficiency of searching electronic legal databases. The topology of American law can be compared to that of other legal systems to determine whether legal systems share universal architectural features, and in what respects different systems are unique. Changing dynamics of the citation frequency and the fitness of particular cases can be studied over historical periods to test historiographical hypotheses. So, for example, Farber\u27s hypothesis that changes in constitutional interpretation occur suddenly, and many others, may be tested rigorously. The dynamics of authority in law generally can be studied much more rigorously. The mere fact that law is a scale free, not a random network, suggests a high degree of intellectual coherence, contrary to what some critics have suggested. The shape of the degree distribution graph of the Web of Law, in its similarity to the scientific citation network, also suggests that cases age, in the sense of losing the ability to attract citations, over time, just as scientific papers do. Yet Supreme Court cases seem to age more slowly. How nodes age profoundly affects overall network structure and therefore affects the shape of the Web of Law. Network theory hints at complex, but analyzable, interactions between the legal doctrines of precedent, and the systems of common law and multiple sovereignties. Because law grows and because it has doctrines of authority, it creates a network of a certain shape, which spontaneously organizes itself. This is the product of laws that govern networks of computers as inexorably as they govern networks of cases, laws arising from the underlying mathematics of networks. Legal databases, which are huge, precisely documented, and readily accessible, present a perfect opportunity for the application of network science. This research would produce new knowledge of general jurisprudence that has simply been impossible until now, when we have the necessary advances in network science, the fast computers, and the existence of a complete record of the legal network in electronic form, waiting to be explored

Algorithms For Discovering Communities In Complex Networks

It has been observed that real-world random networks like the WWW, Internet, social networks, citation networks, etc., organize themselves into closely-knit groups that are locally dense and globally sparse. These closely-knit groups are termed communities. Nodes within a community are similar in some aspect. For example in a WWW network, communities might consist of web pages that share similar contents. Mining these communities facilitates better understanding of their evolution and topology, and is of great theoretical and commercial significance. Community related research has focused on two main problems: community discovery and community identification. Community discovery is the problem of extracting all the communities in a given network, whereas community identification is the problem of identifying the community, to which, a given set of nodes belong. We make a comparative study of various existing community-discovery algorithms. We then propose a new algorithm based on bibliographic metrics, which addresses the drawbacks in existing approaches. Bibliographic metrics are used to study similarities between publications in a citation network. Our algorithm classifies nodes in the network based on the similarity of their neighborhoods. One of the drawbacks of the current community-discovery algorithms is their computational complexity. These algorithms do not scale up to the enormous size of the real-world networks. We propose a hash-table-based technique that helps us compute the bibliometric similarity between nodes in O(m ?) time. Here m is the number of edges in the graph and ?, the largest degree. Next, we investigate different centrality metrics. Centrality metrics are used to portray the importance of a node in the network. We propose an algorithm that utilizes centrality metrics of the nodes to compute the importance of the edges in the network. Removal of the edges in ascending order of their importance breaks the network into components, each of which represent a community. We compare the performance of the algorithm on synthetic networks with a known community structure using several centrality metrics. Performance was measured as the percentage of nodes that were correctly classified. As an illustration, we model the ucf.edu domain as a web graph and analyze the changes in its properties like densification power law, edge density, degree distribution, diameter, etc., over a five-year period. Our results show super-linear growth in the number of edges with time. We observe (and explain) that despite the increase in average degree of the nodes, the edge density decreases with time

A topological space for design, participation and production. Tracking spaces of transformation

'Space of transformation' is a concept borrowed from Serres' communication theory and here redefined after the evolution of the post-digital milieu and the materialistic critique of the same. Hackerspaces, fablabs, medialabs and other shared machines shops are defined here as spaces of transformation, places for the encounter between humans and non-humans, where disciplines are bridged together, hitherto severed, giving place to collective practices related to education, production and society. Shared machine shops are sited locally but also connected globally. Online, they share innovative forms of production, education and collective organization, giving place to a complex ecosystem. This article presents analysis of the topology of this ecosystem conducted by means of tracking and visualizing the online interactions between the hackerspaces listed at the platform Hackerspaces.org. The application of network analysis is aimed to answer the research questions: First, how shared machine shops are locally and globally connected? Second, what links hackerspaces among them and these with new social issues? The concept of shared machine shops as spaces of transformation and the study of their mutual relations allows for an understanding of the transformative capacity of these spaces and how they are producing a new space for social innovation through its mutual interchange of information

A survey of statistical network models

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference